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Standard Deviation Calculator: Mean, Median, and Five-Number Summary
Paste one list of numbers to compute mean, median, mode, variance, standard deviation, five-number summary, quartiles, IQR, range, and coefficient of variation, with a sample or population toggle.
Standard deviation
Paste a list
Your descriptive summary appears here after you calculate.
Mean
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sample s = sqrt( sum( (x - mean)^2 ) / (n - 1) )
How?
How this is calculated
Every statistic comes from one pasted list. The mean is the sum divided by the count. The median is the middle order statistic, or the average of the two middle values when the count is even. The mode is every value tied at the highest frequency, so a list can be bimodal or multimodal, and a list with no repeat reports "none (no value repeats)".
Variance is the sum of squared deviations from the mean, divided by the sample divisor `n - 1` by default or the population divisor `n` when you switch the basis. Standard deviation is the square root of variance. The coefficient of variation is the standard deviation divided by the mean, shown as a ratio and as a percent, following scipy's variation semantics.
Quartiles and the five-number summary use Hyndman-Fan type-7 linear interpolation: for probability p the position is h = (n - 1) p, and the quartile is interpolated between the two nearest order statistics. IQR is Q3 minus Q1. Sample variance, standard deviation, and CV are undefined at n = 1, and CV is undefined when the mean is 0, so those read as "Not defined" rather than 0, Infinity, or NaN. Full precision is kept in exports; the visible values round to four decimals.
Formula: sample s = sqrt( sum( (x - mean)^2 ) / (n - 1) )
Which quartile method, and why it matters
Textbook methods disagree on quartiles for small samples, so a Q1 that looks "wrong" is usually a different convention. This tool uses type-7, the default of the spreadsheets and libraries most people already use. The contrast below is computed on the list 1, 2, 3, 4, 5, 6, 7, 8.
Four quartile methods compared on 1 to 8
| Method | Q1 | Q3 | Where you see it |
|---|---|---|---|
| Type 7 (this tool) | 2.75 | 6.25 | Excel QUARTILE.INC, Sheets, NumPy, R default |
| Tukey hinges | 2.5 | 6.5 | TI-83, Moore & McCabe |
| Type 6 / exclusive | 2.25 | 6.75 | Minitab, Excel QUARTILE.EXC |
| Mendenhall-Sincich | 2 | 7 | some intro textbooks |
The grouping above holds only for even counts. On an odd list like 1 to 9, type-7 gives Q1 3 and Q3 7, and Tukey's hinges coincide with it at 3 and 7 because Tukey includes the median in both halves when the count is odd. Only the TI-83 and Moore & McCabe method, which excludes the median, gives 2.5 and 7.5 there.
Sample or population, which to use
Use the population divisor `n` when your list is the whole group you care about. Use the sample divisor `n - 1` when the list is a sample and you want to estimate the spread of the wider population it came from. This tool defaults to the sample divisor because a pasted list is almost always a sample.
The default basis differs by context across this suite. Here the default is the sample divisor `n - 1`, whereas the z-score tool defaults to the population divisor `n` because standardization treats the pasted list itself as the reference distribution. The same list yields sample SD 2.1381 here and population SD 2 there, each default answering its own question.
Sources
- NIST SEMATECH e-Handbook, Quantitative Techniques (measures of location and scale). NIST. Retrieved .
- Standard deviation. Wikipedia. Retrieved .
- The Many Meanings of Quartile (locating quartiles). The Math Doctors. Retrieved .
- scipy.stats.variation (coefficient of variation). SciPy. Retrieved .